The topological snake lemma and corona algebras
type: Item
instance of: scholarly article
title: The topological snake lemma and corona algebras
zbMATH Open document ID: 0932.46045
published in: The New York Journal of Mathematics
publication date: 1999-10-10T00:00:00Z
full work available at URL: https://eudml.org/doc/120151
full work available at URL: 5-11.html
review text: If \(B\) is a \(C^*\)-algebra then the multiplier algebra of \(B\) is denoted by \({\mathcal M}B\) and the Corona algebra is denoted by \({\mathcal Q}B={\mathcal M}B/B\). A *-isomorphism \(f: B\to B'\) is quasi-unital when there is a projection \(p\in{\mathcal M}B\) such that the closed linear span of \(f(B)B'\) has the form \(pB'\). The main result of the paper is the following. Suppose that \(B\) and \(B'\) are separable \(C^*\)-algebras and that \(f:B\to B'\) is a quasi-unital monomorphism, then the induced map \(\overline f:{\mathcal Q}B\to{\mathcal Q}B'\) is a monomorphism, too. This result is based on the new \(C^*\)-algebra version of the so-named snake lemma from homological algebra.
Mathematics Subject Classification ID: 46L05
Mathematics Subject Classification ID: 18G35
Mathematics Subject Classification ID: 22A05
Mathematics Subject Classification ID: 46L85
Mathematics Subject Classification ID: 46M18
zbMATH DE Number: 1348672
zbMATH Keywords: multiplier algebras
zbMATH Keywords: Corona algebra
zbMATH Keywords: separable \(C^*\)-algebras
zbMATH Keywords: quasi-unital monomorphism
zbMATH Keywords: snake lemma
zbMATH Keywords: homological algebra
author: Claude L. Schochet
reviewed by: V. M. Deundjak
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1348672